Fractal Analysis of Shear-thinning Fluid Flow through Porous Media

Abstract

The fractal capillary models for calculating the volumetric flow rates and permeabilities for Newtonian, power-law, Ellis and Bingham fluids in packed beds are developed by considering fractal nature of the tortuous capillary. The fractal permeability models for Newtonian and non-Newtonian fluids are found to be a function of the tortuosity fractal dimension, the pore-area fractal dimension, sizes of particles and clusters, the effective porosity and the flow behavior of a non-Newtonian fluid. The volumetric flow rate of each fluid as a function of pressure drop are calculated from both the converging-diverging duct approach and the derived expressions in order to compare two models with one another. In addition, hydraulic conductivity is also obtained in terms of the fractal scaling parameters. The volumetric flow rates of shear-thinning fluids, including power-law and Ellis fluids decrease with increasing the tortuosity fractal dimension. It is found that the fractal capillary model for the Newtonian and the Ellis fluids is in good agreement with the converging-diverging duct approach for the considered values of the tortuosity fractal dimension.

Keywords

• Shear-thinning fluid,Porous media,Packed bed,Permeability,Fractal modeling

References

1. 1. Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2008). Transport phenomena, Revised 2nd Edition, Wiley, New York.
2. 2. Sabiri, N. and Comiti, J. (1995). Pressure drop in non-Newtonian purely viscous fluid flow through porous media. Chem. Eng. Sci., 50: 1193-1201.
3. 3. Eslami, A. and Taghavi, S.M. (2017). Viscous fingering regimes in elasto-visco-plastic fluids. J. Non-Newt. Fluid Mech., 243: 79–94.
4. 4. Tijani, H.I., Abdullah, N., Yuzir, A. and Ujang, Z. (2015). Rheological and fractal hydrodynamics of aerobic granules. Bioresource Techn., 186: 276–285.
5. 5. Shokri, H., Kayhani, M. H. and Norouzi, M. (2017). Nonlinear simulation and linear stability analysis of viscous fingering instability of viscoelastic liquids. Physics of Fluids, 29: 033101 http://dx.doi.org/10.1063/ 1.4977443
6. 6. Balhoff, M.T. and Thompson, K.E. (2006). A macroscopic model for shear-thinning flow in packed beds based on netweork modeling. Chem. Eng. Sci., 61: 698-719.
7. 7. Chhabra, R.P., Comiti, J. and Machac, I. (2001). Flow of non-Newtonian fluids in fixed and fluidized beds. Chem. Eng. Sci., 56: 1-27.
8. 8. Yu, B. and Liu, W. (2004). Fractal analysis of permeability for porous media. AIChE J., 50: 46-57.

9. 9. Yu, B. and Cheng, P. (2002). A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transfer, 45: 2983-2993.
10. 10. Pearson, J.R.A. and Tardy, P.M.J. (2002). Models for flow of non-Newtonian and complex fluids through porous media. J. Non-Newt. Fluid Mech., 102: 447-473.
11. 11. Bird, R.B., Armstrong, R.C. and Hassager, O. (1987). Dynamics of polymeric liquids, Wiley, New York.
12. 12. Comiti, J. and Renaud, M. (1989). A new model for determining mean structure parameters of fixed beds from pressure drop measurements: Application to beds packed with parallel pipedal particles. Chem. Eng. Sci., 44: 1539-1545.
13. 13. Dharamadhikari, R.V. and Kale, D.D. (1985). Flows of non-Newtonian fluids through porous media, Chem. Eng. Sci., 40: 527-529.
14. 14. Yu, B.M. (2005). Fractal character for tortuous stream tubes in porous media. Chinese Physics Letters, 22:158-160.